simplifying nasty complex math expression

I know 100% that this expression must be real since the excitation to the system was real in my Linear, time-invariant system of differential equations.

Somehow, the solutions manual transforms the equation into some e^(jw)s, and uses Euler's identity from there to reach two real sinusoidal functions. The problem I'm having is making the e^(jw) out of that expression in the first place.

Ok, you have some exponentiation going on there, twice. What I would do is convert the rectangular representations you have into polar ones via the following: if you have

, then

, where

, and

.

Exponentiation, and multiplication for that matter, is easier in the polar representation of complex numbers than in the rectangular. However, I would definitely use rectangular for addition and subtraction. So, I would get those numbers into polar, do the exponentiation, convert back to rectangular, and see what you get.